An asymptotic approach to solidification shrinkage-induced macrosegregation in the continuous casting of binary alloys

The modelling of macrosegregation in the continuous casting of alloys normally requires resource-intensive computational fluid dynamics (CFD). By contrast, here we develop an asymptotic framework for the case when macrosegregation is driven by solidification shrinkage; as a first step, a binary alloy is considered. Systematic asymptotic analysis of the steady-state two-dimensional mass, momentum, heat and solute conservation equations in terms of the shrinkage parameter indicates that the overall problem can be reduced to a hierarchy of decoupled problems: a leading-order problem that is non-linear, and a sequence of linear problems, with the actual macrosegregation of the solute then being determined by means of one-dimensional quadrature. A numerical method that solves this sequence is then developed and implemented, and yields realistic macrosegregation profiles at low computational cost.